10 ANALYTICAL MECHANICS move c until its origin falls on the terminus of b, and so on until all the vectors are joined. This gives, in general, an open polygon. Then the resultant is obtained by drawing a vector which closes the polygon and which has its origin at the origin of a. The validity of this method will be seen from Fig. 11, where r represents the resultant vector. Evidently the resultant vanishes when the given vectors form a closed polygon. Second: draw a system of rectangular coordinate axes; resolve each vector into components along the axes; add the components along each axis geometrically, beginning at the origin. This gives the components of the required vector. Then draw the rectangular parallelopiped determined by these components. The resultant is a vector which has the origin of the axes for its origin and forms a diagonal of the parallelopiped.* This method is based upon the following analytical method. 15. Analytical Method. —Expressing the given veetorsand their resultant in terms of their rectangular components, we have Fiu. 11. a= ;i) r = rx + ry + r Substituting from (1) in the vector equation r=-a+b+c + • and collecting the terms we obtain , . But since the directions of the coordinate axes are imlepen- * When the given vectors are in the same plane the panUlelopipeHl mlumn to a rectangle.